Wavelet Numerical Simulation Of Biomembrane Shapes

Lipid membranes is the main body of cell surface,which is the important structural basis of cell life activities.Therefore,lipid vesicles composed of lipid bilayers are often used as simplified models of cells.Despite its simple structure,lipid bilayers can form a variety of different shapes.In fact,in the case of spherical topology,the various shapes observed and their transformations can be well understood as curvature models,the classical curvature model being the Helfrich’s spontaneous curvature elasticity model.Shape equations derived from Helfrich’s spontaneous curvature elasticity model are fourth-order nonlinear differential equations.For lipid membranes with an edge,there are three boundary conditions for nonlinear equations describing the shape of the free edge of lipid membranes.Therefore,there is no way to find general solutions to the shape equations of biomembrane,only some special analytical solutions can be found.Or you can solve it numerically.Based on this background,the exploration of numerical simulation means of biomembrane can not only provide theoretical explanation and prediction for the various shapes of lipid vesicles and their transformation,but also contribute to the artificial and reasonable design of model membranes for life science research.In order to achieve this goal,the key is to deeply understand the types of problems of biomembrane shape equation,and use the corresponding numerical method with high numerical accuracy and stable solution to solve it.Since the thickness of biomembrane is far less than its transverse scale,biomembrane can be treated as a curved surface mathematically.Assuming that the curved surface is smooth enough,then the free energy can be variated by surface variational theory based on external differentiation and moving frame method,and the corresponding shape equation of biomembrane can be derived.Because the surface is smooth enough and has natural geometric boundary conditions,it presents a nonlinear boundary value problem with the shape equation corresponding to the lipid vesicle and a nonlinear initial value problem with the shape equation corresponding to the lipid membrane with an edge.Wavelet numerical method has excellent performance in dealing with nonlinear boundary value problems and nonlinear initial value problems.Therefore,in this paper,a wavelet numerical simulation scheme suitable for axisymmetric lipid vesicles and lipid membranes with an edge is derived by using the wavelet integral collocation method based on Coiflet wavelet.The influence of surface tension,spontaneous curvature and osmotic pressure on the size of dumbbell neck end is discussed in detail,and the shape of lipid membranes with an edge is predicted based on the derived scheme.The main innovative research achievements are as follows:(1)Based on the wavelet integral collocation method,a wavelet numerical scheme was established to simulate the shape of axisymmetric lipid vesicles.The shape equations of spherical solution and butterfly solution were solved.The feasibility of the wavelet numerical scheme was verified by observing the shape of the lipid vesicle.Subsequently,spherical shape was taken as the initial shape,and the effect of reduced osmotic pressure on the shape of lipid vesicles was considered.The results showed that the shape of lipid vesicles evolved from spherical shape to rod shape and then to dumbbell shape with the increase of osmotic pressure.(2)Based on the wavelet numerical scheme to simulate the shape of lipid vesicles,the influence of integral constants in the shape equation on the shape of lipid vesicles is discussed.The results show that the same or similar integral constants should be selected when studying the shape evolution of the same lipid vesicles.Then study the lipid vesicle adhesion on the basal plane shape evolution;Finally,the influence of surface tension,spontaneous curvature of membrane and osmotic pressure of the lipid vesicle on the shape of dumbbell shaped lipid vesicles was explored.The numerical results showed that no matter what kind of physical factors caused the changes in the size of the neck,the free energy of dumbbell shaped lipid vesicles was relatively large when the neck radius of dumbbell shaped lipid vesicles was small.(3)In order to solve the initial value problem which consists of the governing equations of axisymmetric lipid membranes with an edge and their geometric boundary conditions,we establish a wavelet numerical scheme based on the wavelet integral collocation method,and use this numerical scheme to simulate the cup shaped membrane which is consistent with the observed membrane shape in the experiment.Further,the shape of lipid membranes with an edge are predicted,and the numerical results show that in the double-layer coupled area difference model,when the reduced area difference is less than 1,there may be solutions other than cup shaped membranes,for example,the single opening dumbbell shape.Through the numerical simulation of lipid vesicles and lipid membranes with an edge,the validity of the wavelet numerical scheme for solving the shape equation of axisymmetric lipid vesicles and lipid membranes with an edge based on the wavelet integral collocation method was proved.It shows the advantage of the wavelet integral collocation method and provides an effective means to analyze and study the colorful shapes of biomembrane.